I don't think Charles III himself will have much importance: when he died in 1527 OTL, he was already 37. In the Renaissance, Princes generally lived up to their 60s: it thus makes Charles III die around the 1550s. At the time, the Religious Wars had not really started in France: they started mainly after the death of Henri II and the reign of his sons if I'm not wrong.
On the other hand, what might be important is wether or not Charles III is able to father a son that will live on and have sons himself: if so, they would be in line for the succession to the French throne before the Bourbon-Vendôme bloodline that gave us OTL Henri IV and, thus, the Royal House of Bourbon. This could thus result in an ATL Royal House of Bourbon that descends from Duke Charles III: I find this pretty ironic considering the latter betrayed Francis I...
Assuming Charles III does succeed in having that son, we have a few problems to consider:
1) Charles III is a traitor to the French crown. Unless he, his sons or any of his male heirs are pardonned, they can't go back to France. If so, if the Valois die as per OTL, people might have trouble accepting the heirs of a traitor on the French throne: traitor that might have even become a foreigner in the eyes of some.
2) The Religious behavior of Charles III and his heirs. Given the fact he betrayed Francis I to serve under Charles V, it's quite likely that the bloodline stays Catholic: having an heir of Charles III of Bourbon go Protestant isn't unlikely but I think the possibility is rather low. A question then remains: what kind of Catholic would they be? Radicals like the Guises? Or Moderates like the late Valois?
In the first case, it probably means you find Charles III's heirs fighting on the Catholic League's side and, if they access the French throne, Protestant blood will flow in the streets... In the second case, the bloodline of Charles III will more likely seek a compromise between both parties and, if it gets on the throne, the situation would be more or less the same as OTL.
3) Wether or not the cousinage of Charles III's bloodline will cause trouble. OTL Henri IV met problems not only because of his Protestantism but also because he was a relatively distant cousin of Henri III as Saint Louis was their only common ancestor. It would be the same with Charles III and his bloodline, although it can be argued they are closer cousins than OTL Henri IV. It depends on how much that argument weighs (probably not much) and of the ambitions of possible counter-claimants.